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Full text of "Treatise On Analysis Vol-Ii"

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(15.4.9)    Let G be a separable, metrizable, locally compact group. Then the
mapping fju-+ (pi) = (ft)* (denoted by p) is an involution on the Banach
algebra M(G) (15.1.7); this follows from ( Also we have ||ju|| =
||/Z|| = \\fi\\ (13.20.1), hence M(G) is a Banach algebra with involution. It can
be shown that it is not necessarily a star algebra.

(15.4.10)    It is clear that every self-adjoint subalgebra (resp. self-adjoint
closed subalgebra) of a normed algebra with involution (resp. of a star
algebra) is a normed algebra with involution (resp. a star algebra).

(1 5.4.1 1 ) In an algebra A with involution, an element x is said to be self-
adjoint (or hermitiari) if x* = x (the terminology is inspired by the example
(15.4.7)). For each x e A, the elements x = %(x + x*) and x2 = (x  x*)f2i
are self-adjoint; so are xx* and x*x, and in general these two products are
distinct. The element x is said to be normal if xx* = x*x, or equivalently if the
self-adjoint elements jq and x2 commute. If A has a unit element e, then
x e A is said to be unitary if xx* = x*x = e, that is if x is invertible and
x'1 = x*. The unitary elements of A form a multiplicative group; for if x is
unitary, then so is x"1 because (15.4.3) (x"1)* = (x*)"1 = x = (x"1)"1, and
if x, y are unitary then so is xy9 because (xy)* = y*x* = y~"1x~1 = (xy)"1.

(15.4.12) Let A be a Banach algebra with involution, having a unit ele-
ment e T6 0.

(i) For each xeA, Sp(x*) is the image of Sp(x) under the map-
ping^ I

(ii) If A is a star algebra, then for each self-adjoint element x e A we have
Sp(x) c: R, and for each unitary element y e A we have Sp(y) c U.

(i) If x  & is invertible, then so is (x - (e)* = x*  Z,e ((15.4.1) and
(15.4.2)). Hence the result.

(ii) Suppose that a 4- i(l e Sp(x), where a and ft are real. Then, for each
real number A, the complex number a + /(/? + X) belongs to Sp(x + ite) and
therefore, by (15.2.4),

a2 + OS + I)2 g ||* + tie\\ 2 = \\x*x + A2?|| g ||jt**|| + A2 = ||x||2 + A2,
so that

Since this is true for all real numbers A, it follows that /? = 0. On the other
hand, if y is unitary, we have ||^||2 = ||>>*y|| = \\e\\ = 1, hence ||j|| = 1;
similarly By"1!! = 1, and therefore Sp(y) c U (15.2.6).0 such that